Quivers, perverse sheaves, and quantized enveloping algebras.(English)Zbl 0738.17011
In earlier work [J. Am. Math. Soc. 3, 447-498 (1990;Zbl 0703.17008) and Prog. Theor. Phys. 102, Suppl., 175-202 (1990;Zbl 0776.17012)] the author constructed a very remarkable so-called canonical basis for \(U^ -_ q\) where \(U_ q\) is the quantized enveloping algebra for a semisimple Lie algebra of simply laced type. In fact, he had two different methods for this construction, an elementary and a geometric method.
In this paper he extends the geometric method (using representations of quivers, perverse sheaves etc.) to the case of quantum groups associated with arbitrary Kac-Moody algebras. Among the amazing properties of the canonical basis of \(U_ q\) are its integrality properties, positivity properties and the fact that it gives canonical bases for all the integrable simple modules. All these properties are proved to hold also in the general case (in fact even stronger positivity results than in earlier work are obtained).
One of the key ingredients in the proofs is an imitation of the methods from the author’s theory of character sheaves.
In this paper he extends the geometric method (using representations of quivers, perverse sheaves etc.) to the case of quantum groups associated with arbitrary Kac-Moody algebras. Among the amazing properties of the canonical basis of \(U_ q\) are its integrality properties, positivity properties and the fact that it gives canonical bases for all the integrable simple modules. All these properties are proved to hold also in the general case (in fact even stronger positivity results than in earlier work are obtained).
One of the key ingredients in the proofs is an imitation of the methods from the author’s theory of character sheaves.
Reviewer: H.H.Andersen (Aarhus)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20G05 | Representation theory for linear algebraic groups |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Cite
Full Text:DOI
References:
| [1] | A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5 – 171 (French). ·Zbl 0536.14011 |
| [2] | Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137 – 252 (French). ·Zbl 0456.14014 |
| [3] | Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. ·Zbl 0332.16015 ·doi:10.1090/memo/0173 |
| [4] | Victor G. Kac, Infinite-dimensional Lie algebras, Progress in Mathematics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 1983. An introduction. ·Zbl 0537.17001 |
| [5] | Masaki Kashiwara and Pierre Schapira, Microlocal study of sheaves, Astérisque 128 (1985), 235 (English, with French summary). Corrections to this article can be found in Astérisque No. 130, p. 209. ·Zbl 0589.32019 |
| [6] | P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665 – 683. ·Zbl 0671.53045 |
| [7] | George Lusztig, Character sheaves. I, Adv. in Math. 56 (1985), no. 3, 193 – 237. ·Zbl 0586.20018 ·doi:10.1016/0001-8708(85)90034-9 |
| [8] | G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447 – 498. ·Zbl 0703.17008 |
| [9] | -, Canonical bases arising from quantized enveloping algebras, II, Progr. Theor. Phys. 102 (1990). ·Zbl 0703.17008 |
| [10] | R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423 – 432. ·Zbl 0311.14001 ·doi:10.2307/1971080 |
| [11] | Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583 – 591. ·Zbl 0735.16009 ·doi:10.1007/BF01231516 |
| [12] | A. Schofield, Notes on constructing Lie algebras from finite-dimensional algebras, preprint. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
